Question: A car travels the 120 miles from $A$ to $B$ at 60 miles per hour, and then returns to $A$ on the same road. If the average rate of the round trip is 45 miles per hour, what is the rate, in miles per hour, of the car traveling back from $B$ to $A$?
Solution: Let $d$ denote the number of miles in the distance from $A$ to $B$ and let $r$ denote the speed of the car (in miles per hour) on the return trip.  It takes $d/60$ hours to travel from $A$ to $B$ and $d/r$ hours to travel from $B$ to $A$.  Round trip, $2d$ miles are covered in $d/60+d/r$ hours for an average speed of  \[
\frac{2d}{\frac{d}{60}+\frac{d}{r}} \cdot \frac{\frac{60}{d}}{\frac{60}{d}} =
\frac{120}{1+\frac{60}{r}}
\] Setting this expression equal to $45$, we find $r=\boxed{36}$.